Eww on the engagement note and schmoopie being shoved down the kids' throats. BTDT.
As for custody, please - he will not get full. Like others said, it's a threat...and generally an empty one. Not many new lovebirds REALLY want the kids around all the time. I mean, kids are a 'drag' with having to parent and stuff! Doesn't leave much time for dating.
I'm really NOT okay with kids missing things because it's inconvenient; that drives me nuts. I also cannot stand the 'saving homework for mom time' or crap like getting laundry dropped off. XH was guilty of that all, but has gotten better over time and I did talk to him about it some too, it didn't just happen because he got a clue.
One thing we do is that he keeps kids overnights on Sundays (on his weekends) and gets them to school Mondays. That was huge in not getting the kids at say 7pm on a Sunday with 2 hours of homework, plus having to make dinner and do wash before Monday morning...not to mention dealing with the sometimes fallout of a 'weekend at dad's' where there's been chaos or emotions running high and such. It's hard on the kids and you. Something to consider, even if you're not in love with the idea. It's a relief to know I'm not gonna be wondering what's up at 8:30 on a Sunday night - what's in store for me now when kids are tired and still have stuff to do for school.
We're not as organized like keeping a shared calendar, but we do keep each other in loop and when stuff does overlap, we're generally pretty good at helping each other so kids don't miss events like sports or a b-day party.
I only ask to be filled in on out of state overnights; I don't need to know about day trips (as you won't want to divulge everything you do with them either, you know?)
Sadly, OP's aren't generally banned unless they're total fucktards...so you'll have to get used to that. I think in time, we all just sort of find a groove?
I would totally make the kids' schoolwork and their activities as part of your 'fight' with him. Let go of the small stuff, if you will. It doesn't seem small to us - but think of end goal; kids happy and as normal as can be.
Accept that skank will be around - so concentrate on the real stuff. Even if we hate skank, as long as she's good to the kids and life is okay when they're at dad's, that's all we can hope for.